Euclidean distance of set of orthogonal vectors

Question

Let's define $x$ as a vector in $\mathbb R^n$

Let's define $V$ as the set of all vectors orthogonal to $x$, i.e $V$={$y$ in $\mathbb R^n$|$x·y=0$}

Let's define $z$ as another vector in $\mathbb R^n$

Calculate the distance between $z$ and the nearest point to $z$ in $V$, i.e min||z-y|| for a vector $y$ in $V$.

After thinking about this, would the answer be $0$? For example, let's say x is the z-axis (0,0,1). So the vectors in V would be the ones around it of any length in the x-y axis. If z is any other vector in $\mathbb R^n$, wouldn't the euclidean distance between z and a vector in V be $0$? Because you could find any vector in V that would intersect z or be infinitesimally close.

If my thinking process is wrong, any help would be great! I'm looking for a way to formalize my thoughts better :)

Answer 1

Are you familiar with orthogonal decomposition? If we take a vector x as you have done, then we could normalize it and obtain an orthonormal basis with $\frac{x}{||x||}$ as a first vector. Your V would then be the space created by all the other vectors in the orthonormal basis. Also, what do u really mean by "intersection"? If what you defined as $z$ has a non-zero component in the $\frac{x}{||x||}$ axis, then your minimum will be bigger than zero. If you have any questions, feel free to ask

Edit: The correct generalization of the distance from a given vector to the given subspace is the projection of the given vector onto the subspace. The projection can be found by, for example, switching your vector into a basis where the some vectors are the ones that create the subspace and thus looking at the components of the vector in the remaining subspaces.

Answer 2

In your example you chose that $x=(0,0,1)\Rightarrow y=(x,y,0)$ where $x,y\in\mathbb{R}$. If you choose that $z=(1,1,1)$ in your example, you will have that $\min|z-y|=\min|(1-x,1-y,1)|=1$.